Discontinuous Functions

We have had a lot to say so far about continuous functions, but what about discontinuous functions? Is there anything meaningful we might say about them? In order that we might ask more precise questions, let's fix our notation. Throughout this section, we will be concerned with a function f : ℝ → ℝ, and we will write D(f) for the set of points at which f is discontinuous. The questions are: What can we say about D(f)? What kind of set is it? Can any set be realized as the set of discontinuities of a function, or does D(f) have some distinguishing characteristics? To get us started, let's recall a few examples.

Examples 9.1

1. (a) If f is monotone, then D(f) is countable. Conversely, any countable set is the set of discontinuities for some monotone f (see Exercise 2.34).

2. (b)There are examples of functions f, g with D(f) = ℚ and D(g) = ℝ. (What are they?)

In particular, we might ask whether D(f) can be a proper, uncountable subset of ℝ. For example, is there an f with D(f) = ℝ\ℚ? or with D(f) = Δ? The answer to the first question is: No, and to the second: Yes, but to understand this will require a bit of machinery.